So today, the High Court ruled that the Parliament has to approve the British exit of the European Union. From what I see, this is their possibility to prevent Brexit and remain in the European Union - which would be a comprehensible step, now that - after the decision - the British seem to understand what they have done. As far as I know, a lot of the people who voted for the Brexit would now vote against it.

Why is this the case? If, even in unofficial polls, the resistance is big after the referendum, and the original decision was that close (52% towards Brexit), why won't the Parliament arrange a new referendum?

Many would likely see this a breach of democratic trust and with more than a third of Labour supporters believed to have backed Brexit, Labour MPs could risk alienating their constituents if they voted to ignore the vote to leave the EU.

(emphasis mine)

So, basically, if the MPs try to continue to disregard the results, voters can just vote for parties who will regard the result in the next election. It wouldn't be politically wise to try that out.

If Parliament does win the right on whether to trigger Brexit, would lawmakers – who are mainly pro-EU – void the referendum? Probably not. That's because, Murkens says, too many of them come from constituencies whose voters backed Brexit.

Why won't the Parliament arrange a new referendum?

This won't work.

Firstly, with the results 48% vs 52%, there are quite a substantial number of people voting for Brexit. Almost half the country voted to remain and the other half voted to leave.

Yes, if they call a second referendum, 'Remain' might win. However, even if 'Remain' wins by 60% of the vote, you would have to consider the 40% who voted to 'Leave'.

Those who voted for 'Leave' will again try to have a third referendum. Some might challenge the legitimacy of the 2nd referendum. If it continues, the problem just won't be solved.

The British government did respond to a parliamentary petition:

The referendum was “one of the biggest democratic exercises in British history with over 33 million people having their say, the decision must be respected.”

The UK Government has said it will appeal to the Supreme Court and it appears that the key question is about the notion of irreversibility. The thinking being that invoking Article 50 is an irrevocable action and so Parliament must speak on it.

It is entirely possible that the Supreme Court would push the question to the European Court of Justice (it is, after all, a question about EU law). The irony of pro-Brexit folk wanting the European Court to rule against Parliamentary sovereignty has not been lost.

Assuming the decision does need to go to Parliament there is a few givens.

Firstly, it is inconceivable that the Conservative party won't be whipped to vote
to invoke Article 50. There has been strong rhetoric about Brexit meaning Brexit and it would be bizarre in the extreme if the Government didn't require its members to vote to invoke. There will, though, be some dissenters.

Secondly it is highly likely that the Liberals and the Scottish National Party will not vote to invoke. The former because it's very pro-EU, the latter because Scotland voted to remain.

So that leaves the Labour party. It is very hard to say whether they will whip or give a free vote. If whipped it will almost definitely to invoke Article 50 based on a will of the people argument. A free vote may be called if there is expected to be a large number of dissenters.

Whatever way it goes it is likely, though not certain, that the Commons will vote to invoke Article 50.

And then it will go to the House of Lords. Since the Conservative win in May last year, the Lords have voted against important Government legislation a number of times either causing it to be pulled or requiring significant amendment.

The Lords can't actually stop a Bill going through if the Government wants it, there are mechanisms to effectively force things. However, it can make life exceedingly difficult. This raises the curious prospect of an unelected chamber delaying or obstructing a law that is based on a direct popular referendum. I can't even begin to imagine the constitutional ramifications that that raises.

Seems that if the EU would move a bit on less immigration the mess could be resolved. Germany seems willing to take in any immigrates that Great Britain declines.
– TomOSep 6 '17 at 18:46

@TomO for the UK, the problem isn't immigration from outside the EU, it's the principle of free movement of people within the EU. To do anything significant about the latter it would need all EU countries, not just the EU, to agree. Politically, that will never happen. It would be political suicide in Poland, Romania, Bulgaria etc.
– AlexSep 7 '17 at 8:59